Exam 2: Propositional Logic: Syntax and Semantic

Instructions: For 11-20, use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, provide a consistent valuation. -P ≡ (Q • ∼R) P ⊃∼Q (∼P • Q) ⊃ S

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -(A ≡ ∼B) ⊃ ∼(B • A)

use the following key to determine which English sentence best represents the given formula of PL. A: Peirce studied logic. B: James was a pluralist. C: Dewey wrote about thirdness. D: Dewey denigrated the quest for certainty. E: Peirce emphasized education. -∼(C $\lor$ D)

use the following key to determine which English sentence best represents the given formula of PL. A: Peirce studied logic. B: James was a pluralist. C: Dewey wrote about thirdness. D: Dewey denigrated the quest for certainty. E: Peirce emphasized education. -D $\lor$ ∼E

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. Justify your answers by appeal to the meanings of those terms. -[(Q ⊃ R) ⊃ (R ⊃ S)] ≡ ∼(∼Q $\lor$ S)

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.) -(G • H) ≡ I G ≡ ∼J J • H (I $\lor$ K) ⊃ G

Instructions: For questions 11-20, construct complete a truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers. -D • ∼E and ∼(E ⊃ ∼D)

use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, choose an option with a consistent valuation. (There may be other consistent valuations.) -D ≡ (A • B) D $\lor$ (∼E • F) ∼E ⊃ A F ⊃ B A $\lor$ B ∼A $\lor$ ∼B

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. Justify your answers by appeal to the meanings of those terms. -[N ? (O ? P)] ? (N ? O)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -R ⊃ S V ⊃ W X ⊃ (S • ∼W) / (R • V) ≡ X

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -D ⊃∼E ∼E ⊃ F / ∼F ⊃ D

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -P ≡ ∼Q ∼(Q $\lor$ ∼P) Q ⊃∼Q / P

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(A ≡ B) • (A $\lor$ C) C / B

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -M ⊃ N O ⊃ M ∼N M ⊃ ∼O / M

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. -D • ∼E and ∼(E ⊃ ∼D)

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the α assignment; the second variable in the formula read left to right (if any) gets the β assignment; the third variable in the formula read left to right (if any) gets the γ assignment; and the fourth variable in the formula read left to right (if any) gets the δ assignment. For exercises with only one propositional variable, the standard assignment is: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions. -[(E ? ?F) $\lor$ (G ? ?H)] $\lor$ [(?E • ?F) • (?G • ?H)]

Instructions: For 11-20, use indirect truth tables to determine, for each given set of propositions, whether it is consistent. If the set is consistent, provide a consistent valuation. -X ≡ Y Y ≡ ∼Z Z ≡ ∼X

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -X $\lor$ Z X ⊃ (Y $\lor$ Z) ∼Z ⊃ Y / Y

use the following key to determine which of the translations of the given English argument to PL is best. B: Brouwer is an intuitionist. F: Frege is a logicist. G: Gödel is a platonist. H: Hilbert is a formalist. -If Gödel is a platonist, then Frege is a logicist. If Frege is a logicist, Brouwer being an intuitionist is a sufficient condition for Hilbert being a formalist. Gödel is a platonist. Gödel is a platonist if Hilbert is a formalist. Therefore, Gödel is a platonist if Brouwer is an intuitionist.

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the α assignment; the second variable in the formula read left to right (if any) gets the β assignment; the third variable in the formula read left to right (if any) gets the γ assignment; and the fourth variable in the formula read left to right (if any) gets the δ assignment. For exercises with only one propositional variable, the standard assignment is: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \beta \gamma \delta 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 Complete truth tables for each of the following propositions. -U ≡ (Z ⊃ U)